FACULTY OF MATHEMATICS AND PHYSICS
CHARLES UNIVERSITY PRAGUE
KAREL RYCHLÍK (1885 – 1968)
Summary of Ph.D. Thesis
Author: Mgr. Magdalena Hykšová
Advisor: doc. RNDr. Jindřich Bečvář, CSc.
Branch: M8 – General Problems of Mathematics and Informatics
Prague, 2002
MATEMATICKO-FYZIKÁLNÍ FAKULTA
UNIVERZITY KARLOVY V PRAZE
KAREL RYCHLÍK (1885 – 1968)
Autoreferát doktorské disertační práce
Vypracovala: Mgr. Magdalena Hykšová
Školitel: doc. RNDr. Jindřich Bečvář, CSc.
Obor: M8 – Obecné otázky matematiky a informatiky
Praha, 2002
Výsledky tvořící disertaci byly získány během studia na MFF UK v Praze v letech1996–2002.
Doktorand: Mgr. Magdalena Hykšová
Školitel: doc. RNDr. Jindřich Bečvář, CSc.Matematický ústav UKSokolovská 83, 186 75 Praha 8
Školící pracoviště: Matematický ústav UK
Oponenti: prof. RNDr. Ladislav Procházka, DrSc.Sekaninova 20, 120 00 Praha 2
prof. RNDr. Štefan Schwabik, DrSc.Matematický ústav AV ČRŽitná 20, 120 00 Praha 2
Obhajoba disertační práce se koná dne v hodinpřed komisí pro obhajoby disertačních prací v oboru M8 – Obecné otázky ma-tematiky a informatiky na MFF UK, Ke Karlovu 3, Praha 2, v místnosti .
S disertací je možno se seznámit na Útvaru doktorandského studia MFF UK,Ke Karlovu 3, Praha 2.
Předseda oborové rady M8: doc. RNDr. Jindřich Bečvář, CSc.Matematický ústav UKSokolovská 83, 186 75 Praha 8
Autoreferát rozeslán dne:
CONTENTS
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Life of Karel Rychlík . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3 Work of Karel Rychlík . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.1 Survey of Scientific Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Algebra and Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2.1 g-adic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.2 Valuation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.3 Theory of Algebraic Numbers, Abstract Algebra . . . . . . . . . . . . . . . . . . . . 143.2.4 Determinant Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.1 The List of Publications of Karel Rychlík . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Obr. 1 Karel Rychlík
1 INTRODUCTION
The thesis is devoted to the life and work of the Czech mathematician Karel Rychlíkand it consists of eight parts. The first of them contains the brief sketch of Rychlík’s life,the survey of his scientific and pedagogical activities and the detailed description of hislife story. The following five chapters discuss particular groups of Rychlík’s publications:2. Algebra and Number theory, 3. Works on Mathematical Analysis, 4. Textbooks, Popu-larizing Papers, Translations, 4. Karel Rychlík and Bernard Bolzano, 5. Other Works onHistory of Mathematics. These chapters are conceived separately, each of them is providedwith the conclusion and the list of references.
The seventh chapter presents the list of Rychlík’s publications, reviews and lectu-res at Charles University, at the Czech Technical University and in the Union of CzechMathematicians and Physicists. The thesis ends with the pictorial appendix, the surveyof abbreviations and the name index.
The aim of this abstract is to provide the basic information on Rychlík’s life andon his most important mathematical results. Part 3 contains the survey of Rychlík’sscientific activities, including their brief evaluation. From all groups of publicatios onlythe principal works are described in separate parts, namely the papers devoted to g-adicnumbers, valuation theory, algebraic number theory and abstract algebra, determinanttheory. More details concerning particular groups of Rychlík’s publications can be found(in addition to the thesis) in the couple of author’s papers [H6] and [H7] and on Internetpages devoted to Rychlík: http://euler.fd.cvut.cz/publikace/HTM/Index.html. For thesake of lucidity, the appendix contains the complete list of Rychlík’s publications and thelist of cited literature.
2 LIFE OF KAREL RYCHLÍK
Karel Rychlík was born on April 16, 1885 in Benešov near Prague as the first of thethree children of Barbora Srbová, married Rychlíková (1865 – 1928), and Vilém EvženRychlík (1857 – 1923).1 In October 1904 he started to study mathematics and physics atthe Faculty of Arts of Czech Charles-Ferdinand University in Prague (below only CharlesUniversity). He was influenced most of all by Professor Karel Petr. In the school year1907/08 Rychlík was studying at Faculté des Sciences in Paris. He was mainly interested inthe lectures of Jacques Hadamard (winter semester) and Émile Picard (summer semester)called Analyse supérieure. Besides, Rychlík attended the lectures of Gaston Darboux,Edouard Goursat, Louis Raffy, Paul Painlevé and Marie Curie at the same faculty andthe lectures on number theory at Collège de France, read by Georges Humbert. During his
1Karel Rychlík had a younger brother Vilém (1887 – 1913), who was a brilliant mathematician, too.Karel used to say his brother had been much cleverer than him. It is a stumper because Vilém died veryyoung, at the age of 26. He had just finished the study of mathematics and physics at the Faculty of Artsof Charles University, received the degree Doctor of Philosophy, become an assistant at Czech TechnicalUniversity in Prague and he had written several treatises. He is told being very lively, loving women andsmoking 40 cigarettes a day, which became fateful for him. One day he caught a cold somewhere andwithin three days he died (that was called a fast consumption).Their younger sister Jana studied, as an adjunct student, mathematics and biology at the Faculty ofArts and became a biology teacher. But soon she married Václav Špála, later the famous Czech painter,and gave precedence to her husband and children.
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stay in Paris Rychlík also worked on his dissertation. On December 16, 1908 he passed theso-called ”teacher examination”. On March 30, 1909, on the grounds of the dissertationand rigorosum examinations of mathematics and philosophy, he was awarded the degreeDoctor of Philosophy.
From 1909 till 1913 Rychlík worked as an assistant of the mathematical seminar at theFaculty of Arts of Charles University. In 1912 he was appointed associate professor (Do-cent). As a ”private associate professor” (this position was not paid in general) Rychlíklectured at the university till 1938. In 1919 the board of professors decided on his appoint-ment adjunct professor, in addition to the present chairs, but their suggestion remainedin the ministry and was not put into practice (the financial situation of the school systemwas not very good). In the end Rychlík became a professor (adjunct: November 27, 1920;full: March 12, 1924) at the Czech Technical University in Prague (below only TechnicalUniversity), where he had been working as an assistant since 1913. In October 1914 heundertook the duties of Professor František Velísek, who had enlisted and died in the war.Rychlík began to read base lectures alternately for students of the first and second yearof study, the lecture on probability theory and the lecture on vector analysis.
From today’s view, it was a pity that Rychlík remained only private associate pro-fessor at Charles University. The main subject of his research was algebra and numbertheory. It was possible, even necessary, to read such topics at Charles University. In fact,Rychlík was the first who introduced methods and concepts of ”modern” abstract algebrain our country - by means of his published treatises as well as university lectures. Besides,as a professor there he would have had a stronger influence on the young generation ofCzech mathematicians. But Rychlík spent most of his time (and energy) at the TechnicalUniversity where he had to adapt his lectures for future engineers. Nevertheless, he ap-proached his work seriously there. In addition to the usual teaching activities, he was amember of many committees, such as organization committee, inceptive committees, etc.
In 1904 Rychlík became a member of the Union of Czech Mathematicians and Physi-cists (below only the Union) and until World War II he was also a member of its committee.Almost the whole of his life Rychlík lectured in the Union and his lectures were very clo-sely related to his scientific research. He was also a member of the Royal Bohemian Societyof Sciences (elected on January 11, 1922), the Czech Academy of Sciences and Arts (May23, 1924) and the Czechoslovak National Research Council under the Academy (May 19,1925). Besides, Rychlík took part in several congresses: 5th Congress of Czech Naturalistsand Physicians in Prague (1914; contribution [R11]), International Congress of Mathema-ticians in Strassbourg (1920), 6th Congress of Czechoslovak Naturalists, Physicians andEngineers in Prague (1928), International Congress of Mathematicians in Bologna (1928;contrib. [R28]), Congress of Mathematicians of Slavonic Countries in Warszawa (1929;contrib. [R33]) and Second Congress of Mathematicians of Slavonic Countries in Prague(1934; contrib. [R42]). In 1939 all Czech universities were closed, after the war Rychlíkretired. In the last period of his life Rychlík invested his energy to the history of mathe-matics, above all to the inheritance of Bernard Bolzano, which he had been interested insince his youth, but after the retirement he was engaged in this topic fully. Karel Rychlíkdied on May 28, 1968 at the age of 83.
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3 WORK OF KAREL RYCHLÍK
3.1 Survey of Scientific Activities
Rychlík’s works can be divided into five groups:
1. Algebra and Number Theory (22 works),
2. Mathematical Analysis (7),
3. Textbooks, Popularization Papers and Translations (16),
4. Works Devoted to Bernard Bolzano (14),
5. Other Works on History of Mathematics (29).
In the Czech mathematical community, Rychlík’s name is mostly related to his text-books on elementary number theory ( [R37], [R46] ) and on the theory of polynomials withreal coefficients [R64], which are certainly very interesting and useful, but which are not”real” scientific contributions. Worth mentioning is the less known textbook [R44] (1938)on probability theory, written for students of the technical university, yet in a very topicalway: Rychlík builds the probability theory using the axiomatic method that is similar tothe one of Kolmogorov [12]. In this context, let us also mention the popularization papers[R5] and [R6] on the special cases (n = 3, 4, 5) of Fermat last theorem, which are cited inRibenboim’s book [26].
Not only among mathematicians and not only in Bohemia, Rychlík is widely knownas the historian of mathematics, above all in the connection with Bernard Bolzano. Asfar as the number of citations is concerned, this domain is unequivocally in the firstplace. Preparing for printing Bolzano’s manuscript Functionenlehre [R34] and two partsof Zahlenlehre ( [R36], [R85] ), Rychlík earned the place in practically all Bolzano’s biblio-graphies. Well-known is also Rychlík’s paper [R19] containing the proof of continuityand non-differentiability of so-called Bolzano’s function. Besides, in the literature devotedto Bernard Bolzano we can often find citations of Rychlík’s papers concerning Bolzanoslogic [R68] and the theory of real numbers ( [R65], [R66] ) that were based on the study ofBolzano’s manuscripts. It should be emphasized that Rychlík sooner than the others mademany of important suprises hidden in Bolzano’s hardly readable manuscripts accessibleand hence contributed to Bolzano’s fame in the mathematical community.
A range of other papers on the history of mathematics more or less relates to Bolzanotoo, namely the works devoted to N. H. Abel [R88], A.-L. Cauchy ( [R58], [R59], [R60],[R61], [R69], [R86] ) and the prize of the Royal Bohemian Society of Sciences for theproblem of the solution of any algebraic equation of a degree higher than four in radicals( [R81], [R82] ). Some of the remaining papers are only short reports ([R44], [R53], [R54],[R61]) or loose processing of literature ([R75], [R77]), the others contain a good deal ofan original work based on primary sources, namely the papers devoted to É. Galois [R62],F. Korálek [R79], M. Lerch ([R27], [R73]), E. Noether [R70], F. Rádl ([R55], [R56]), B.Tichánek ([R25], [R74], [R78]), E. W. Tschirnhaus [R76] and F. Velísek [R20]. Moreover,Rychlík adds his own views and valuable observations, which shows his wide insight anddeep interest in the history of mathematics and in mathematics itself. On October 21,1968 the Czechoslovak Academy of Sciences awarded Rychlík in memoriam a prize forthe series of 13 papers on the history of mathematics published after 1957, namely [R57],[R58], [R59], [R62], [R64], [R66], [R67], [R68], [R70], [R76], [R77], [R81] and [R87].
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Rychlík’s algebraic works are known only to a relatively narrow circle of mathemati-cians. But it is just this first group, in which the most important mathematical papersof Karel Rychlík are included. These works, discussed in the part 3.2, were publishedbetween 1914 and 1932, that is in the period of the birth and formation of ”modern” abs-tract algebra, and they were devoted to particularly topical problems from this domain.Regrettably only three papers of Karel Rychlík were published in a generally reputablemagazine – Crelle’s Journal; the most of them were published in de facto local Bohemianjournals. It was certainly meritorious for enlightenment in the Czech mathematical audi-ence, but although some of the works were written in German, they were not noticed bythe mathematical community abroad, even though they were referred in Jahrbuch überdie Fortschritte der Mathematik or Zentralblatt für Mathematik und ihre Grenzgebiete(nevertheless, it was not only Rychlík who published mostly for the Czech audience; infact, this situation was common at that time in the young autonomous republic). On theother hand, Rychlík’s papers published in Crelle’s Journal became known and they havebeen cited in the literature – this concerns above all the treatise Zur Bewertungstheorieder algebraischen Körper [R22] from 1923, thanks to which Rychlík gained a certain po-sition in the history of valuation theory. This paper is cited for example by R. Böffgena M. A. Reichert ([1], 1987), H. Hasse ([4], 1926; [6], 1933), A. N. Kochubei ([11], 1998),W. Krull ([13], 1930; [14], 1932), M. Nagata ([17], 1953), W. Narkiewicz ([18], 1974),A. Ostrowski ([20], 1933; [21], 1935), P. Ribenboim ([24], 1985), P. Roquette ([27], 1999),O. F. G. Schilling ([28], 1950), F. K. Schmidt ([6], 1933; [29], 1933), W. Wiȩs law ([30],1988) and others.
In his papers Rychlík mostly came out of a certain work (see Fig. 3) and gave someimprovements – mainly he based definitions of the main concepts or proofs of the maintheorems on another base, in the spirit of abstract algebra, and in this way he generalizedor simplified them. Typical features of the papers are the brevity, conciseness, topicalityas well as the ”modern” way of writing (from the point of view of that time). Althoughthe amount of these publications is not very large and they are relatively short, theygive evidence of Rychlík’s wide horizons, of the fact that he followed the current worldmathematical literature, noticed problems and possible generalizations that later provedto be substantial, aimed for correct but as simple as possible proofs.
A bit aside stand Rychlík’s occasional works on mathematical analysis. The papers[R17] and [R21] devoted to continuous non-differentiable functions in p-adic number fieldsare closely related to the previous group of Rychlík’s publications. It is interesting thatthis couple of papers represents one of the first works studying p-adic analysis at all.
It shall be added that even in the citation database Web of Science, which monitors8440 ”valuable” journals from 1980 to the present, it is possible to find eleven citationsof Rychlík’s publications (twice the paper on valuation theory [R22], otherwise worksconcerning Bernard Bolzano: five times [R85], once [R34], [R19], [R82] and [R86] ). Ne-vertheless, the total amount of citations after the year 1980 is greater than eleven – thereare other citations in books as well as in journals and proceedings that are not monitoredby the database.
Figure 1 demonstrates the distribution of subjects of Rychlík’s interest, their develo-pment and changes in the course of time, as well as connections of publications to otherscientific activities. We can observe:
1 At the beginning of his career Rychlík wrote several works on algebra without a
5
1910 - 21: editor of ČPMF - tasks for students
1960
19501950
1940
1930
1920
1910
7
2
3
1
[R11], 1914g-adic numbers
Algebra and Number Theory
Popularizing Papers,Textbooks, Translations
[R12], 1916g-adic numbers
[R15], [R16], 1919Divisibility theory
[R23], [R24], 1923Divisibility theory
[R22], 1923Valuation theory
[R14], 1919Valuation theory
[R31], [R32], 1929Congruences
[R26], 1924Ideal theory
[R39], [R40], 1932Artin theorem
[R1], 1907Interpolation th.[R2], 1908
Equations[R3], 1909Group of transf.
[R4], 1910[R7], 1911
Algebraic forms
[R5], [R6], 1910Fermat last th. [R8], 1911
Continued fractions[R9], 1912Regular 17-gon
[R18], 1921Quadratic fields
Translations from Russian
Mathematical Analysis
[R17], 1920[R21], 1923
A continuous non-
diferent. function in
g-adic number field
[R10], 1912Power series
[R13], 1917Summation of series
[R41], [R42], 1933Nonregular sequences
[R33], 1929Sochocki's method
[R37], 1931Elementary number
theory (textbook)
[R38], 1931Determinant theory
[R43], 1934-35 Determinant theory [R44], 1938
Probability theory
(textbook)
[R46], 1950Elementary number
theory (textbook)
[R47], 1950, Glivenko Probability theory
[R48], 1952, Chinčin Continued fractions
[R49], 1953, Kuroš Alg. equations[R50], 1955, Tichonov,
Samarski, Equations of
Math. Physics[R64], 1957, Theory of polynomials (textbook)
[R70], 1958Diophant. equat.
[R72], 1958Number theory
[R83], 1960Pythagoras theorem
U1912-30
C1914
M1907
M1909
M1906
M1918 M1919, 1922
M1924
M1931
C1934
C1929
M1913
4
5
U1931/32
6
T1914-39
U1931-37
Fig. 1 Survey of Scintific Activities of Karel Rychlík
6
8
9
Works Devoted
to Bernard BolzanoOther Works on the History of Mathematics
[R19], 1922
Bolzano's function
[R20], 1922
F. Velísek[R25], 1923
Computing of e
[R27], 1925
M. Lerch[R28], 1928
Bolzano's theory of functions
[R34], 1930
Functionenlehre
[R36], 1931
Zahlentheorie
[R45], 1942-43
J. F. Kulik
[R62], [R71], 1957, 58, E. Noether
[R51], [R65], [R66],
1956, 57, 58,
Bolzano's theory
of real numbers
[R54]-[R57],
1957, F. Rádl
[R58]-[R61], 1957
[R69], 1958
A. L. Cauchy
[R52], [R53],
1957, Auto-
biografie
[R63], 1957
E. Galois
[R67], [R68], 1958
Bolzano's logic[R73], 1959
Bolzano's stay
in Liběchov
[R74], 1959
M. Lerch
[R75], 1959
[R79], 1960
Tichánek
[R76], 1959
Arab.num.
[R77], 1959
Tschirnhaus
[R78], 1959
Bourbaki
[R80], 1960, F. Korálek [R81], 1960
Preisaufgabe..[R84], [R85], 1961,
1962, Bolzano's
th. of real numbers
[R86], 1962
Bolzano & Cauchy[R88], 1964
N. H. Abel
1924: Bolzano committee
M1959
C1928
M1927
M1930
M1920
Lecture in the UnionM1959
C1928Legend: Lecture at a congress(in 1928)
Regular lecture at the Czech Tech. Univ.
Regular lecture at Charles UniversityU1912-30
T1914-39
[R35], 1930
Bolzano, B.
10
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deeper relation to his later publications. We only note that [R3] concerning substitutiongroups was a dissertation, [R4] and [R7] on the theory of algebraic forms were inceptiveworks.
2 In 1910 – 1921 Rychlík was an editor of the magazine Časopis pro pěstování mate-matiky a fyziky with the responsibility for tasks for secondary school students, publishedin the supplement of this journal. In the same time five Rychlík’s popularizing papers( [R5], [R6], [R8], [R9], [R18] ) appeared in this supplement.
3 The most important mathematical papers by Karel Rychlík concern algebra andnumber theory and they originated mainly by the middle of the twenties. Precisely, by theyear 1924 when the Bolzano Committee under the Royal Bohemian Society of Scienceswas established - compare 8 . Rychlík was its member since the very beginning – and itshould be emphasized that he was a very active member. Soon (Fig. 2) we will see thatthe most of the remaining papers involved in the group 3 have their origin – or at leastinspiration – in those published earlier. Rychlík’s lectures at Charles University in 1912 -1930 were closely related just to this domain, similarly with the lectures in the Union.
4 Seven papers belong to mathematical analysis. Among them there are two couples[R17], [R21] and [R41], [R42]) consisting of Czech and German variant of almost the sametext. Otherwise, the works on analysis are mutually independent.
Although the list of publications itself leads to the opinion that mathematical analysiswas only a marginal domain of Rychlík’s interest, the list of lectures shows that it wasthe main domain of his pedagogical duties at the Technical University. Consequently, inaddition to the activities in the Bolzano Committee, it was another reason why Rychlíkdid not publish more algebraic papers.
5 An interesting example of mutual relations of publication activities and lecturingis related to probability theory. In the school year 1914/15 Rychlík started reading the”classical lecture named Probability calculus at the Technical University. In the summersemester of the school year 1931/32 Rychlík lectured on Probability calculus (the theoryof Mises) at Charles University. For the winter semester of the school year 1933/34 Rych-lík had announced the lecture on linear algebra, but shortly before the beginning of thesemester he changed the topic for Probability calculus (from the axiomatic point of view);hence we can see that he immediately reacted to the publication of Kolmogorov’s book[12], the first work where the probability theory was built axiomatically. In 1938 Rychlík’stextbook Introduction to probability calculus [R44] was published. Although it was inten-ded for students of the Technical University, it was written in a very topical way, usingKolmogorov’s axiomatic method. As far as we can judge by the textbook, the quality ofRychlík’s lectures in the period before the World War II was outstanding. Towards theend of working on the textbook, in the winter semester of the school year 1936/37, Rychlíklectured on Probability calculus from the axiomatic point of view at the Charles Universityone more time. As it was outlined above, within the domain of probability theory we canobserve the development from the classical lecture at the technical university over thestudy of modern trends and their modifications and improvements up to the publicationof a well-elaborated textbook.
A close relation to the probability theory can also be found in the couple of Rychlík’spapers [R41] and [R42] published in 1933, which are included in the group of publicationson mathematical analysis. Rychlík came back to probability theory once again at the endof the World War II, when he started to translate Glivenko’s book Probability theory;the translation [R47] was published in 1950.
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6 World War II, postwar years and the beginning of a communist epoch were hard forKarel Rychlík. He did not return to the Technical nor to Charles University, the BolzanoCommittee was abolished together with the Royal Bohemian Society of Sciences in 1951.By the middle of the fifties only a short paper [R45] on the history of mathematics, thesecond edition [R46] of a former textbook [R37] on elementary number theory and fourtranslations from Russian – see 7 – had been published.
8 The situation became less unpleasant in 1955, when the ”new” CzechoslovakAcademy of Sciences officially entrusted Rychlík with organizing Bolzano’s manuscriptinheritance (during the following years he was sometimes given a reward for a particularwork). In the same year the Czech Literary Fund, which supported old scientists andtheir widows, started to pay him a regular remuneration. Moreover, in 1958 the BolzanoCommittee was restored under the academy (nevertheless only for three years). A glimpseat the Figure 1 is sufficient to notice that after many unfortunate years a fertile periodsuddenly came.
9 Besides the manuscripts of Bernard Bolzano, Rychlík was deeply interested in thehistory of mathematics in general. The most of his historical papers were devoted to acertain personality; remaining four papers are included in the group denoted by 10 .
3.2 Algebra and Number Theory
Rychlík’s papers on algebra and number theory can be divided as follows.
Principal Papers
g-adic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [R11], [R12], [R17], [R21]Valuation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [R14], [R22]Algebraic Numbers, Abstract Algebra . . . . . . . . . . . . . . [R15], [R16], [R23], [R24], [R26]
[R31], [R32], [R33], [R39], [R40]Determinant Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [R38], [R43]
Other Works
Theory of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [R2]Theory of Algebraic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [R4], [R7]Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [R3]
Figure 2 illustrates the influences in the development of the algebraic number theory.The aim of the scheme is to show Rychlík’s place there; hence there are not all existinginfluences – the predetermination of the figure would be covered up. It just tries to showthe two main streams, the ideal theory represented by Richard Dedekind and his continua-tors, and the divisor theory represented by Leopold Kronecker, his student Kurt Hensel,his student Helmut Hasse and other mathematicians, including Karel Rychlík.
The survey of quotations in Rychlík’s principal algebraic papers (except the two paperson determinant theory that stay a little bit aside) is given in Figure 3. It is evident thatRychlík was above all influenced by Kurt Hensel.
9
C. F. Gauss (1777 – 1855)
C. G. J. Jacobi (1804 – 1851)
F. G. M. Eisenstein (1823 – 1852)
P. G. L. Dirichlet (1805 – 1859)
J. Liouville (1809 – 1882)
G. Lamé (1795 – 1871)
. . . . . .
?E. E. Kummer (1810 – 1893)
R. J. W. Dedekind
(1831 – 1916)
?
L. Kronecker
(1823 – 1891)
?
E. I. Zolotarev
(1847 – 1878)
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J. V. Sochocki
(1842 – 1927)
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K. Hensel
(1861 – 1941)
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D. Hilbert (1862 – 1943)
E. Noether (1882 – 1935)
E. Artin (1898 – 1962)
P. Furtwängler (1869 – 1940)
T. Takagi (1875 – 1960)
. . . . . .
E. Steinitz
(1871 – 1928)
?J. Kürschák
(1864 – 1933)
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´´́
´́+K. Rychlík
(1885 – 1968)-
¾
À
A. Ostrowski
(1893 – 1986)
½½
½=½½
½>
H. Hasse (1898 – 1979)
W. Krull (1899 – 1970)
O. F. G. Schilling (1911 – 1973)
B. L. van der Waerden (1903 – 1996)
I. Kaplansky, S. MacLane, M. Nagata,
W. Narkiewicz, P. Ribenboim,
P. Roquette, F. K. Schmidt, W. Wieslaw,
. . . . . .
Fig. 2 Some Influences in the Development of Algebraic Number Theory
10
E. Steinitz, 1910Algebraische Theorie...
[R12], 1916On Hensel's Numbers
[R15], 1919The Divisibility in Alg.
Number Fields
[R16], 1920Divis. Th. of Alg. Numbers...
[R17], 1920A Continuous Nowhere Diff. Fun. in
the Field of Hensel's Numbers
[R21], 1923Eine stetige nicht diff. Funktion im Gebiete der Henselschen Zahlen
[R23], 1923Zur Theorie der Teilbarkeit
[R24], 1923Zur Theorie der Teilbarkeit in alg.
Zahlkörpern
[R26], 1924Eine Bemerkung zur Theorie der Ideale [R31], 1929
Ext. Congr. for Alg. Number f.
[R22], 1923Zur Bewertungstheorie...
[R14], 1919A Contrib. to the Field Theory
[R32], 1929On the Ext. of Congruence
K. Hensel, 1908Theorie der alg. Zahlen
J. Kürschák, 1913Über Limesbildung...
K. Hensel, 1913Zahlentheorie J. Sochocki, 1893
Zasada najwiekszego...
[R11], 1914A Remark on Hensel's Th. of Alg. Numbers
A. Ostrowski, 1918Über einige Lösungen ...
E. I. Zolotarev, 1882 Sur la théorie des nombres...
[R38], 1932On the Artin Theorem
[R39], 1932Über d. Artin. Verfeinerungssatz
[R33], 1929Über die Anwendung die Methode von Sochocki
Fig. 3 Survey of Citations in Rychlík’s Principal Papers
3.2.1 g-adic Numbers
In the first paper [R11] of the considered group Rychlík generalizes Hensel’s ideasconcerning additive and multiplicative normal form of g-adic numbers, which he extendsto algebraic number fields.
The second paper [R12] is devoted the introduction and properties of the ring ofg-adic numbers. While Hensel took the way analogous to the construction of the field ofreal numbers by means of decimal expansions, Rychlík came out – alike Cantor – fromthe concepts of fundamental sequence and limit. As he notes, one of the merits of thisapproach is, that directly from the definition, it can be immediately seen that the ring ofg-adic numbers depends only on primes contained in g, not on their powers. Of course,the idea of constructing the field of p-adic numbers (for a prime p) came from Kürschák[15], who introduced the concept of valuation. Rychlík generalized the notion of limitin a slightly different way, closer to Hensel. Moreover, he studied comprehensively ringsof g-adic numbers for a composite number g. Kürschák’s paper [15] is cited only in thepostscript that seems to be written subsequently. It is plausible he came to the idea of
11
the generalization of Cantor’s approach independently of Kürschák.2 In the mentionedpostscript Rychlík generalized Kürschák’s technique for the case of the composite numberg and defined what was later called pseudo-valuation of a ring R. 3
In 1920 Karel Petr published in the Czech journal Časopis pro pěstování mathematikya fysiky a very simple example of a continuous non-differentiable function [22]. Only theknowledge of the definition of continuity and derivative and a simple arithmetic theoremis necessary to understand both the construction and the proof of continuity and non–differentiability of the function. Petr’s function is defined in the interval [0, 1] as follows:
x =a1
101+
a2
102+
a3
103+
a4
104+ · · · ; ak ∈ {0, 1, . . . , 9}; (1)
f(x) =b1
21±
b2
22±
b3
23±
b4
24± · · · ; bk = {
0 for even ak,1 for odd ak,
(2)
the sign before bk+1 is opposite than the one before bk for ak ∈ {1, 3, 5, 7}, the sameotherwise.
The graph of an approximation of Petr’s function can be seen in the left picture below.To show it more graphically, a four-adic number system was used. Comparing with thegraph on the right, the necessity of the exception to the rule of sign assignment awardedto the digit 9 can be understood; the result would not be a continuous function:
In the same year and in the same journal Karel Rychlík generalized Petr’s functionin his paper [R17]; the German version [R21] with the same content was published twoyears later in Crelle’s journal. Rychlík carried the function from the real number field Rto the field of p–adic numbers Qp:
x = arpr + ar+1pr+1 + · · · , r ∈ Z, ai ∈ {0, 1, . . . , p − 1}; (3)
f(x) = arpr + ar+2pr+2 + ar+4pr+4 + · · · . (4)
The proof that the function (4) is continuous in Qp, but has a derivative nowhere inthis field, is rather elementary. At the end Rychlík remarks it would be possible to follow
2At least since 1909, when he lectured in the Union On Algebraic Numbers according to Kurt Hensel, Rychlíkhad been involved in this topics and was trying to improve Hensel’s ideas – here the solid foundation of the basicconcepts was in the first place.
3It is almost unknown but interesting that Rychlík defined this concept 20 years before the publication ofMahler’s paper [16], which is usually considered as a work where the general pseudo-valuation (Pseudobewertung)was introduced.
12
the same considerations in any field of p-adic algebraic numbers (introduced by K. Hensel)subsistent to the algebraic number field of a finite degree over Q.
We shall remark that this Rychlík’s work was one of the first published papers dealingwith p-adic continuous functions. In Hensel’s [9] some elementary p-adic analysis can befound, but otherwise it was developed much later.
3.2.2 Valuation Theory
In his paper [15] J. Kürschák introduced the concept of valuation (Bewertung ) as amapping ‖ · ‖ of a given field K into the set of non-negative real numbers, satisfying thefollowing conditions:
‖a‖ > 0 if a ∈ K, a 6= 0; ‖0‖ = 0, (V1)
‖1 + a‖ ≤ 1 + ‖a‖ for all a ∈ K, (V2)
‖ab‖ = ‖a‖ . ‖b‖ for all a, b ∈ K, (V3)
∃a ∈ K : ‖a‖ 6= 0, 1. (V4)
The main result of Kürschák’s paper is the proof of the following theorem.Every valued field K can be extended to a complete, algebraically closed
valued field.First, Kürschák constructs the completion of K in the sense of fundamental sequences;
it is not difficult to extend the valuation from K to its completion. Then he extends thevaluation from the complete field to its algebraic closure. Finally he proves that thecompletion of the algebraic closure is algebraically closed. The most difficult step is thesecond one. Kürschák shows that if α is a root of a monic irreducible polynomial
f(x) = xn + a1xn−1 + · · · + an, ai ∈ K (an = ±Nα), (9)
it is necessary to define its value as ‖α‖ = ‖an‖1
n . To prove that this is the valuation,the most laborious and lengthy point is the verification of the condition (V2). For thispurpose Kürschák generalizes Hadamard’s results concerning power series in the complexnumber field. Nevertheless, at the beginning of his paper Kürschák remarks that in allcases, where instead of the condition (V2) a stronger condition
‖a + b‖ ≤ Max(‖a‖, ‖b‖) for all a, b ∈ K. (V2’)
holds, i.e. for non-archimedean valuations , it is possible to generalize Hensel’s considerati-ons concerning the decomposition of polynomials over Qp, especially the assertion, latercalled Hensel’s Lemma :
If the polynomial (9) is irreducible and ‖an‖ ≤ 1, then it is also ‖ai‖ ≤ 1 forall coefficients ai, 1 ≤ i ≤ n − 1.
He didn’t prove Hensel’s Lemma for a field with a non-archimedean valuation - hewrote he had not succeeded in its generalization for all cases, it means for archimedeanvaluations too. So he turned to the unified proof based on Hadamard’s theorems, validfor all valuations.
A. Ostrowski proved in his paper [19] that every field K with an archimedean valuationis isomorphic to a certain subfield of the complex number field C in the way that for everya ∈ K and the corresponding a ∈ K it is ‖a‖ = |a|ρ, where | · | is the usual absolute value
13
on C, 0 < ρ < 1, ρ does not depend on a (such valuations are called equivalent). In otherwords, up to isomorphism, the only complete fields for an archimedean valuation are Rand C, where the problem of the extension of valuation is trivial. Hence it is possible torestrict the considerations only to non-archimedean valuations and use the generalizationof Hensel’s Lemma.
And precisely this was into full details done by Karel Rychlík in [R14] and [R22]. Thesecond paper is the German variant of the first one written in Czech with practically thesame content. But only the German work became wide known – thanks to its publication inCrelle’s journal, while its Czech original was not noticed by the mathematical communityabroad. The paper [R22] is cited e.g. by R. Böffgen, H. Hasse, A. N. Kochubei, W. Krull,M. Nagata, W. Narkiewicz, A. Ostrowski, P. Ribenboim, P. Roquette, O. F. G. Schilling,F. K. Schmidt and W. Wiȩs law (see page 5). In the connection with some variant of theabove lemma, Rychlík’s name is mentioned also without the explicite citation of the work;see the papers I. Kaplansky [10], J. Eršov [3], I. Efrat and M. Jarden [2], the recent book[25] of P. Ribenboim etc.
3.2.3 Theory of Algebraic Numbers, Abstract Algebra
The papers included in this group were published in Czech journals, in Czech orGerman, and remained almost unknown outside Bohemia. They are, nevertheless, veryinteresting and manifest Rychlík’s wide horizons as well as the fact that he followed thelatest development in the theory, studied the current mathematical literature, noticedproblems or possible generalizations that later turned out to be important. Let us onlymention that in his papers we can find the definition of divisors in algebraic numberfields via a factor group and an external direct product, introduction of divisibility viathe concept of a semi-group and other ideas; more details can be found in [H6].
3.2.4 Determinant Theory
The first of the couple of Rychlík’s papers devoted to determinants [R38], published inCrelle’s journal in 1931, concerns the assertion that the determinant of a matrix A ∈ Kn×n,n > 1, where two rows or columns are identical, is zero. It can be easily proved for thecase that the characteristic of the given field K is not 2. Rychlík cites the book [5] ofH. Hasse, where a completely general proof using the Laplace’s ”Entwicklungssatz” isgiven. Rychlík gives a simple proof of the considered assertion just for the field K ofcharacteristic 2. He steps as follows. Consider the determinant of a matrix X = (xij) aspolynomial over Z in indeterminates xij. If a matrix X∗ has two identical rows (columns),then it is |X∗| = 0 in a ring which arises from Z by adjunction of the elements of X∗;it is also |X∗| ≡ 0 (mod 2). This implies |X∗| = 0 in a ring which arises from a primefield of K by the adjunction of the elements of X∗, hence also in a ring which arises fromK by this adjunction. If A is a matrix with elements of K and with two identical rows(columns), then the determinant |A| is received from |X∗| by substituting the elementsof A for the elements of X∗, so it is |A| = 0.
This Rychlík’s paper didn’t remain completely unknown – it was cited for example byO. Haupt in the third edition of his Einführung in die Algebra I [7].
The second paper [R43] published in 1934 is written in Czech and it comes out ofthe paper [23] of K. Petr, where the determinant theory in is based on the definition ofa determinant as an alternating m-linear form. Rychlík generalizes Petr’s considerations
14
for the case of an arbitrary field K of an arbitrary characteristic. For this purpose it isnecessary to give a suitable definition of an alternating m-linear form (equivalent to Petr’sone for fields of characteristics different from 2).
4 APPENDIX
The abbreviations of magazines used bellow:Bull. = Bulletin internat. Acad. Boheme; ČPM(F) = Časopis pro pěstování math.
(a fysiky); ČMŽ = Čechoslovackij matem. žurnal – Czechoslovak Math. Journal ; Crelle= Journal für die reine und angewandte Math.;MŠ = Matematika ve škole; Pokroky =Pokroky matematiky, fyziky a astronomie; Rozhledy = Rozhledy matematicko–fysikální;Rozpravy = Rozpravy II. tř. České akad. věd a umění; Věstník = Věstník Královskéčeské spol. nauk – Mémoires de la société royale des sciences de Bohême.
4.1 The List of Publications of Karel Rychlík[R1] Poznámky k theorii interpolace, ČPMF 36(1907), 13–44.
[R2] O resolventách se dvěma parametry, Rozpravy 17(1908), Nr. 31, 5 pp.
[R3] O grupě řádu 360, ČPMF 37(1908), 360–379.
[R4] Příspěvek k theorii forem, Rozpravy 19(1910), Nr. 49, 13 pp.
[R5] O poslední větě Fermatově pro n = 4 a n = 3, ČPMF 39(1910), 65–86.
[R6] O poslední větě Fermatově pro n = 5, ČPMF 39(1910), 185–195, 305–317.
[R7] Příspěvek k theorii forem II, Rozpravy 20(1911), Nr. 1, 5 pp.
[R8] Geometrické znázornění řetězc̊u, ČPMF 40(1911), 225–236.
[R9] Sestrojení pravidelného sedmnáctiúhelníku, ČPMF 41(1912), 81–93.
[R10] Příspěvek k teorii potenčních řad o více proměnných, ČPMF 41(1912), 470–477.
[R11] Poznámka k Henselově theorii algebraických čísel, Věstník pátého sjezdu českých přírodozpytc̊uva lékař̊u v Praze, 1914, 234–235.
[R12] O Henselových číslech, Rozpravy 25(1916), Nr. 55, 16 pp.
[R13] O de la Vallée-Poussinově metodě sčítací , ČPMF 46(1917), 313–331.
[R14] Příspěvek k theorii těles, ČPMF 48(1919), 145–165.
[R15] Dělitelnost v alg. tělesech číselných vzhledem k rac. prvočíslu, Rozpravy 28(1919), Nr. 14, 5 pp.
[R16] Theorie dělitelnosti čísel algebraických, Rozpravy 29(1920), Nr. 2, 6 pp.
[R17] Funkce spojité nemající derivace pro žádnou hodnotu proměnné v tělese čísel Henselových, ČPMF49(1920), 222–223.
[R18] O kvadratických tělesech číselných, ČPMF 50 (1921), 49–59, 177–190.
[R19] Über eine Funktion aus Bolzanos handschriftlichem Nachlasse, Věstník 1921–22, Nr. 4, 6 pp.
[R20] Ph. Dr. Frant. Velísek (posmrt. vzpomínka), ČPMF 51(1922), 247–248.
[R21] Eine stetige nicht differenzierbare Funktion im Gebiete der Henselschen Zahlen,Crelle 152(1922–23), 178–179 [German translation of [R17] ].
[R22] Zur Bewertungstheorie der algebraischen Körper, Crelle 153(1923), 94–107.
[R23] Zur Theorie der Teilbarkeit, Věstník 1923, Nr. 5, 32 pp.
[R24] Zur Theorie der Teilbarkeit in algebraischen Zahlkörpern, Věstník 1923, Nr. 9, 36 pp.
[R25] Číselný výpočet čísla e, ČPMF 52(1923), 300.
[R26] Eine Bemerkung zur Theorie der Ideale, Věstník 1924, Nr. 10, 9 pp.
[R27] Seznam vědeckých prací zemř. prof. Matyáše Lercha, ČPMF 54(1925), 140–151 [with K. Čupr].
15
[R28] La Théorie des Fonctions de Bolzano, Atti del Congresso internazionale dei Matematici, Bologna,1928 (publ. 1931), vol. 6, 503–505.
[R29] O Cantorových řadách a zlomcích g-adických, Rozpravy 37(1928), Nr. 2, 6 pp.
[R30] Sur les fractions g-adiques et les séries de Cantor, Bull. 29(1928), 153–155 [French tran. of [R29]].
[R31] O rozšíření pojmu kongruence pro alg. tělesa konečného stupně, Rozpravy 38(1929), Nr. 21, 4pp.
[R32] O rozšíření pojmu kongruence, ČPMF 58(1929), 92–94.
[R33] Über die Anwendung der Methode von Sochocki, Sprawozdania z Pierwszego kongresu matematy-ków Krajów Slowianskich, Warszawa, 1929 (publ. 1930), 181–184.
[R34] B. Bolzano, Functionenlehre, KČSN, Praha, 1930 [edited and notes by Rychlík; forew. by Petr].
[R35] Bolzano, Bernard, in: Ottův slovník naučný nové doby , J. Otto, Praha, 1930, 675.
[R36] B. Bolzano, Zahlentheorie, KČSN, Praha, 1931 [Rychlík edited and provided with notes].
[R37] Úvod do elementární teorie číselné, JČMF, Praha, 1931, 102 pp.
[R38] Eine Bemerkung zur Determinantentheorie, Crelle 167(1931), 197.
[R39] O větě Artinově, Rozpravy 42(1932), Nr. 23, 3 pp.
[R40] Über den Artinschen Verfeinerungssatz, Bull. 33(1932), 149–152 [German transl. of [R39]].
[R41] Poznámka k Böhmerovým nepravidelným posloupnostem, Rozpravy 43 (1933), Nr. 8, 4 pp.
[R42] Bemerkung über Böhmers regellose Folgen, Bull. 34(1933), 15–16 [German transl. of [R41]].
[R43] Determinanty v tělesech libovolné charakteristiky, ČPMF 64(1934–35), 135–140.
[R44] Úvod do počtu pravděpodobnosti, JČMF, Praha, 1938, 144 pp.
[R45] Jakub Filip Kulik, Rozhledy 22(1942–43), 88–89.
[R46] Úvod do elementární číselné theorie, Přírodovědecké nakl., Praha, 1950, 192 pp.
[R47] V. I. Glivenko, Teorie pravděpodobnosti, Přírod. nakl., Praha, 1950, 248 pp. [transl. by Rychlík].
[R48] A. J. Chinčin, Řetězové zlomky, Přírod. nakl., Praha, 1952, 104 pp. [transl. by Rychlík].
[R49] A. G. Kuroš, Algebraické rovnice libovolných stupň̊u, SNTL, Praha, 1953, 42 pp. [tr. by Rychlík].
[R50] A. N. Tichonov - A. A. Samarskij, Rovnice matematické fyziky, Nakl. ČSAV, Praha, 1955, 765 pp.[transl. by K. Rychlík and A. Apfelbeck].
[R51] Teorie reálných čísel v Bolzanově rukopisné poz̊ustalosti, ČPM 81(1956), 391–395.
[R52] Jak jsem studoval matematiku, Praha, 1956, 21 pp. (cyclostyled).
[R53] Jak jsem studoval matematiku, MŠ 7(1957), 300–309 [extract of [R52]].
[R54] Prof. dr. František Rádl, Rozhledy 35(1957), 285.
[R55] Prof. dr. František Rádl, Pokroky 2(1957), 600.
[R56] Profesor dr. František Rádl zemřel, ČPM 82(1957), 378–381 [with L. Rieger].
[R57] Seznam pojednání prof. dr. Františka Rádla, ČPM 82(1957), 381–382.
[R58] Cauchyho rukopis v archivu ČSAV, ČPM 82(1957), 227–228.
[R59] Cauchyho rukopis v archivu ČSAV, Pokroky 2(1957), 633–637 [more detailed variant of [R58]].
[R60] Un manuscrit de Cauchy aux archives de l’Académie tchécoslovaque des sciences, ČMŽ 7(82)(1957), 479–481.
[R61] Un manuscrit de Cauchy aux archives de l’Académie tchécoslovaque des sciences, Revue d’hist.sciences 10(1957), 256–261 [overprint of [R60]].
[R62] K 75. výročí narození Emmy Noetherové, Pokroky 2(1957), 611.
[R63] Évariste Galois, Pokroky 2(1957), 729–733.
[R64] Úvod do analytické teorie mnohočlen̊u s reálnými koeficienty, Nakl. ČSAV, Praha, 1957, 181 pp.
[R65] Theorie der reellen Zahlen im Bolzanos handschriftlichen Nachlasse, ČMŽ7(82)(1957), 553–567.
[R66] Těorija věščestvěnnych čisel v rukopisnom nasledii Bolzano, Istor.–matěm. issledovanija 11(1958),515–532 [Russian transl. of [R65] by A. I. Lapina].
[R67] Úvahy z logiky v Bolzanově rukopisné poz̊ustalosti, ČPM 83(1958), 230–235.
16
[R68] Betrachtungen aus der Logik im Bolzanos handschriftlichen Nachlasse, ČMŽ 8(83)(1958), 197–202[German transl. of [R67]].
[R69] Cauchys Schrift ”Mémoire sur la dispersion de la lumière”, ČMŽ 8(83)(1958), 619–632.
[R70] Diofantická rovnice x3 + y3 + z3 = k, MŠ 8(1958), 22–28.
[R71] Emmy Noetherová – nejvýznamnější žena matematička, MŠ 8(1958), 234–238.
[R72] 1958–19.58=8591–85.91, MŠ 8(1958), 591–597.
[R73] Bolzan̊uv pobyt v Liběchově, MŠ 9(1959), 111–113.
[R74] Mat. Lerch a jeho odpovědi na otázky ankety o metodě práce matematik̊u, MŠ9(1959), 170–173.
[R75] Výpočet čísla e, základu přirozených logaritm̊u, MŠ 9(1959), 394–402.
[R76] P̊uvod ”arabských” číslic, MŠ 9(1959), 553–561.
[R77] K 250. výročí Tschirenhausa, Pokroky 4(1959), 232–234.
[R78] Nicolas Bourbaki, Pokroky 4(1959), 673–678.
[R79] Výpočet základu e přirozených logaritm̊u, ČPM 85(1960), 37–42.
[R80] Matematik Filip Koralek, náš krajan, a jeho pobyt v Paříži v polovině minulého století, Pokroky5(1960), 472–478.
[R81] Úloha o cenu, vypsaná r. 1834 Královskou českou společností nauk k oslavě jejího padesátiletéhotrvání, Zprávy komise ČSAV 4(1960), 24.
[R82] Preisaufgabe der Königlichen böhmischen Gesellschaft der Wissenschaften zu Prag für das Jahr1834, ČPM 86(1961), 76–89 [more detailed variant of [R81]].
[R83] O formulaci Pythagorovy věty, MŠ 12(1961–62), 629–630.
[R84] La théorie des nombres reéles daus un ouvrage posthume manuscrit de B. Bolzano, Revue d’hist.sciences 14(1961), 313–327.
[R85] Theorie der reelen Zahlen in Bolzanos handschrift. Nachlasse, Nakl. ČSAV, Praha, 1962, 103 pp.
[R86] Sur les contacts personnels de Cauchy et de Bolzano, Revue d’hist. sciences 15 (1962), 163–164.
[R87] Volodimir Fomič Bržečka, Pokroky 7(1964), 191–192.
[R88] Niels Henrik Abel a Čechy, Pokroky 7(1964), 317–319.
4.2 References
[1] Böffgen, R.; Reichert, M. A., Computing the decomposition of primes p and p-adic absolute valuesin semisimple algebras over Q, Journal of symbolic computation 4(1987), 3–10.
[2] Efrat, I.; Jarden, M., Free pseudo p-adically closed fields of finite corank, Journal of Algebra133(1990), 132–150.
[3] Eršov, J. L., Multiply valued fields, Doklady akademii nauk SSSR 253(1980), 274–277.
[4] Hasse, H., Über die Einzigkeit der beiden Fundamentalsätze der elementaren Zahlentheorie, Crelle155(1926), 199–220.
[5] Hasse, H., Höhere Algebra I, Sammlung Göschen, Berlin, 1926.
[6] Hasse, H.; Schmidt, F. K., Die Struktur diskret bewerteter Körper, Crelle 170(1933), 4–63.
[7] Haupt, O., Einführung in die Algebra I, II, Akademische Verlagsgesellschaft, Leipzig, 1929; 3. vyd.1956.
[8] Hensel, K., Theorie der algebraischen Zahlen I, B. G. Teubner, Leipzig, 1908.
[9] Hensel, K., Zahlentheorie, G. J. Göschen, Berlin-Leipzig, 1913.
[10] Kaplansky, I., Maximal Fields with Valuations, Duke Math. Journal 9(1942), 303–321.
[11] Kochubei, A. N., Harmonic oscillator in characteristic p, Letters in mathematical physics 45(1998),11–20.
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[12] Kolmogorov, A. N., Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer Verlag, Berlin, 1933.
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[17] Nagata, M., On the Theory of Henselian Rings, Nagoya Math. Journal 5, February (1953), 45–57.
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Magdalena Hykšová – The List of Publications
[H1] Life and Work of Karel Rychlík , in: WDS97, MFF UK, Prague, 1997, 148–152.
[H2] Fraktály a jejich objektově orientované definice [Fractals and Their Object Orien-ted Definitions ], in: Matematika v proměnách věků I, Prometheus, Prague, 1998,192–210.
[H3] Bratři Weyrové a jejich kořeny v našem kraji [Brothers Weyrs and Their Roots inOur Region], in: Někteří významní matematici mající vztah k našemu kraji ], Prácekatedry matematiky 6, listopad 1998, Gaudeamus, Hradec Králové, 1998, 23–32.
[H4] Rodenhausen, A.; Peitgen, H.-O.; Skordev, G.,Self-affine Curves and SequentialMachines, review for Zentralblatt fr̈ Mathematik , r722201, 1998.
[H5] Karel Rychlík a Bernard Bolzano, in: IX. Seminář o filozofických otázkách mate-matiky a fyziky, Prometheus, Prague, 2000, 51–62.
[H6] Life and Work of Karel Rychlík , in:Mathematics throughout the Ages, Prometheus,Prague, 2001, 67–91.
[H7] Bolzano’s Inheritance Research in Bohemia, in: Mathematics throughout the Ages,Prometheus, Prague, 2001, 258–286.
[H8] Fraktály a objektově orientované programování [Fractals and Object Oriented Pro-gramming ], Pokroky matematiky, fyziky a astronomie 46(2001), 232–253.
[H9] Karel Rychlík and His Mathematical Contributions , in: 6. Tagung der FachsektionGeschichte der Mathematik (conference proceedings), to appear, 2002, 10 pages.
[H10] A Methodological Approach to Global Evaluation of the Scientific Work of a Per-sonality , in: V. Österreichisches Symposion zur Geschichte der Mathematik (con-ference proceedings), to appear, 2002, 9 pages.
[H11] Evaluation of Scientific Work – Methodological Notes , in: Die Biographik als Zu-gang zur Wissenschaftsgeschichte (conference proceedings), to appear, 2002, 15pages.
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